We consider a dynamic capillarity equation with stochastic forcing on a compact Riemannian manifold
(
M
,
g
)
(M,g)
:
d
(
u
ε
,
δ
−
δ
Δ
u
ε
,
δ
)
+
d
i
v
f
ε
(
x
,
u
ε
,
δ
)
d
t
=
ε
Δ
u
ε
,
δ
d
t
+
Φ
(
x
,
u
ε
,
δ
)
d
W
t
,
\begin{equation*} d \left (u_{\varepsilon ,\delta } -\delta \Delta u_{\varepsilon ,\delta }\right ) +div\mathfrak {f}_{\varepsilon }(\mathbf {x}, u_{\varepsilon ,\delta })\, dt =\varepsilon \Delta u_{\varepsilon ,\delta }\, dt + \Phi (\mathbf {x}, u_{\varepsilon ,\delta })\, dW_t, \end{equation*}
where
f
ε
\mathfrak {f}_{\varepsilon }
is a sequence of smooth vector fields converging in
L
p
(
M
×
R
)
L^p(M\times \mathbb {R})
(
p
>
2
p>2
) as
ε
↓
0
\varepsilon \downarrow 0
towards a vector field
f
∈
L
p
(
M
;
C
1
(
R
)
)
\mathfrak {f}\in L^p(M;C^1(\mathbb {R}))
, and
W
t
W_t
is a Wiener process defined on a filtered probability space. First, for fixed values of
ε
\varepsilon
and
δ
\delta
, we establish the existence and uniqueness of weak solutions to the Cauchy problem for the above-stated equation. Assuming that
f
\mathfrak {f}
is non-degenerate and that
ε
\varepsilon
and
δ
\delta
tend to zero with
δ
/
ε
2
\delta /\varepsilon ^2
bounded, we show that there exists a subsequence of solutions that strongly converges in
L
ω
,
t
,
x
1
L^1_{\omega ,t,\mathbf {x}}
to a martingale solution of the following stochastic conservation law with discontinuous flux:
d
u
+
d
i
v
f
(
x
,
u
)
d
t
=
Φ
(
u
)
d
W
t
.
\begin{equation*} d u +div\mathfrak {f}(\mathbf {x}, u)\,dt =\Phi (u)\, dW_t. \end{equation*}
The proofs make use of Galerkin approximations, kinetic formulations as well as
H
H
-measures and new velocity averaging results for stochastic continuity equations. The analysis relies on the use of a.s. representations of random variables in some particular quasi-Polish spaces. The convergence framework developed here can be applied to other singular limit problems for stochastic conservation laws.